Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

NDA 2015 · Mathematics · Inverse Trigonometric Functions

If y= ^ -1 [ 1+ x + 1- x 1+ x - 1- x ], where 0 < x < (π)/(2),then...

If y=cot⁡−1[1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x],where0<x<π2,thendydxy=\cot ^{-1} \left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right], where 0 < x < \frac{\pi}{2},then \frac{d y}{d x}is equal to

Options

  1. A1/2
  2. B2
  3. Csin⁡x+cos⁡x\sin x+\cos x
  4. Dsin⁡x−cos⁡x\sin x-\cos x

Correct answer

A. 1/2

Step-by-step solution

y=cot⁡−1[1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x]\begin{aligned} & y=\cot ^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right]\end{aligned}y=cot⁡−1[(cos⁡x2+sin⁡x2)2+(cos⁡x2−sin⁡x2)2(cos⁡x2+sin⁡x2)2−(cos⁡x2−sin⁡x2)2]\begin{aligned} & y=\cot ^{-1}\left[\frac{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^2}+\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^2}}{\sqrt{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^2}-\sqrt{\left(\cos \frac{x}{2}-\sin \frac{x}{2}\right)^2}}\right] \end{aligned}y=cot⁡−1[cos⁡x2+sin⁡x2+cos⁡x2−sin⁡x2cos⁡x2+sin⁡x2−cos⁡x2+sin⁡x2]y=cot⁡−1[2cos⁡x22sin⁡x2]=cot⁡−1(cot⁡x2)=x2dydx=12\begin{aligned} & y=\cot ^{-1}\left[\frac{\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}+\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}}{\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}-\cos \frac{\mathrm{x}}{2}+\sin \frac{\mathrm{x}}{2}}\right]\\ & \mathrm{y}=\cot ^{-1}\left[\frac{2 \cos \frac{\mathrm{x}}{2}}{2 \sin \frac{\mathrm{x}}{2}}\right]=\cot ^{-1}\left(\cot \frac{\mathrm{x}}{2}\right)=\frac{\mathrm{x}}{2} \\ & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}\end{aligned}
Practice Inverse Trigonometric Functions on Quantrex Academy →

More from Inverse Trigonometric Functions

Consider the following in respect of inverse circular functions : I. ^ -1 (-x)=- ^ -1 x… NDA 2026What is [2 ^ -1 (1)/(2)-(π)/(4)] equal to ? NDA 2026If ^ -1 k + ^ -1 (1)/(2) = (π)/(4) , then what is the value of k ? NDA 2025What is the value of [(1)/(2) ^ -1 ( 2 3 )]? NDA 2025Consider the following for the two (02) items that follow: Let y = ^ -1 (x - (4x^3)/(27))… NDA 2025If k is a root of x^2 - 4x + 1 = 0 , then what is ^ -1 k + ^ -1 (1)/(k) equal to? NDA 2025What is ^ -1 9 + cosec^ -1 ( 41 4 ) equal to? NDA 2025What is ^2 ( ^ -1 2)+ ^2 (cosec^ -1 3) equal to? NDA 2024If ^ -1 x= ^ -1 x , then which one of the following is correct? NDA 2024What is 15+ ^2((π)/(4)-2 ^ -1 3) equal to? NDA 2024If 4 ^ -1 x+ ^ -1 x=π , then what is ^ -1 x+4 ^ -1 x equal to? NDA 2024What is ^ -1 ((a)/(b))- ^ -1 ((a-b)/(a+b))equal to? NDA 2024What is 2 ((1)/(2) ^ -1 5 3 )equal to? NDA 2023If ^ -1 p-cosec^ -1 q=0 , where p > 0, q > 0 ; then what is the value of ^ -2 + ^ -2 ? NDA 2023What is 1+ ^2 ( ^ -1 ( 3 17 ))equal to? NDA 2023What is ^ -1 (cosec^ -1 2) equal to? NDA 2022 Full Inverse Trigonometric Functions list Inverse Trigonometric Functions JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs